• Previous Article
    On the nodal set of the eigenfunctions of the Laplace-Beltrami operator for bounded surfaces in $R^3$: A computational approach
  • CPAA Home
  • This Issue
  • Next Article
    Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit
September  2014, 13(5): 2095-2113. doi: 10.3934/cpaa.2014.13.2095

Stability of delay evolution equations with stochastic perturbations

1. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Campus Reina Mercedes, Apdo. de Correos 1160, 41080 Sevilla

2. 

Department of Higher Mathematics, Donetsk State University of Management, Chelyuskintsev str., 163-a, Donetsk, 83015

Received  December 2012 Revised  February 2013 Published  June 2014

The investigation of stability for hereditary systems is often related to the construction of Lyapunov functionals. The general method of Lyapunov functionals construction, which was proposed by V.Kolmanovskii and L.Shaikhet, is used here to investigate the stability of stochastic delay evolution equations, in particular, for stochastic partial differential equations. This method had already been successfully used for functional-differential equations, for difference equations with discrete time, and for difference equations with continuous time. It is shown that the stability conditions obtained for stochastic 2D Navier-Stokes model with delays are essentially better than the known ones.
Citation: Tomás Caraballo, Leonid Shaikhet. Stability of delay evolution equations with stochastic perturbations. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2095-2113. doi: 10.3934/cpaa.2014.13.2095
References:
[1]

T. Caraballo, M. J. Garrido-Atienza and J. Real, Asymptotic stability of nonlinear stochastic evolution equations,, \emph{Stoch. Anal. Appl.}, 21 (2003), 301. doi: 10.1081/SAP-120019288. Google Scholar

[2]

T. Caraballo and K. Liu, Exponential stability of mild solutions of stochastic partial differential equations with delays,, \emph{Stoch. Anal. Appl.}, 15 (1999), 743. doi: 10.1080/07362999908809633. Google Scholar

[3]

T. Caraballo, K. Liu and A. Truman, Stochastic functional partial differential equations: existence, uniqueness and asymptotic decay properties,, \emph{Proc. Roy. Soc. Lond. A}, 456 (2000), 1775. doi: 10.1098/rspa.2000.0586. Google Scholar

[4]

T. Caraballo, J. Real and L. Shaikhet, Method of Lyapunov functionals construction in stability of delay evolution equations,, \emph{J. Math. Anal. Appl.}, 334 (2007), 1130. doi: 10.1016/j.jmaa.2007.01.038. Google Scholar

[5]

H. Chen, Asymptotic behavior of stochastic two-dimensional Navier-Stokes equations with delays,, \emph{Proc. Indian Acad. Sci (Math. Sci)}, 122 (2012), 283. doi: 10.1007/s12044-012-0071-x. Google Scholar

[6]

V. Kolmanovskii and L. Shaikhet, A method of Lyapunov functionals construction for stochastic differential equations of neutral type,, \emph{Differentialniye uravneniya}, 31 (2002), 691. Google Scholar

[7]

V. Kolmanovskii and L. Shaikhet, Construction of Lyapunov functionals for stochastic hereditary systems: a survey of some recent results,, \emph{Mathematical and Computer Modelling}, 36 (1995), 1851. Google Scholar

[8]

J. Luo, Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays,, \emph{J. Math. anal. Appl.}, 342 (2008), 753. doi: 10.1016/j.jmaa.2007.11.019. Google Scholar

[9]

E. Pardoux, Equations aux dérivées partielles stochastiques nonlinéaires monotones,, Ph.D thesis, (1975). Google Scholar

[10]

G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions. Encyclopedia of mathematics and its applications,, Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223. Google Scholar

[11]

L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Difference Equations,, Springer, (2011). doi: 10.1007/978-0-85729-685-6. Google Scholar

[12]

L. Shaikhet, Modern state and development perspectives of Lyapunov functionals method in the stability theory of stochastic hereditary systems,, \emph{Theory of Stochastic Processes}, 2 (1996), 248. Google Scholar

[13]

L. Wan and J. Duan, Exponential stability of non-autonomous stochastic partial differential equations with finite memory,, \emph{Statistics and Probability Letters}, 78 (2008), 490. doi: 10.1016/j.spl.2007.08.003. Google Scholar

[14]

M. Wei and T. Zhang, Exponential stability for stochastic 2D-Navier-Stokes equations with time delay,, \emph{Appl. Math. J. Chinese Univ.}, 24 (2009), 493. Google Scholar

show all references

References:
[1]

T. Caraballo, M. J. Garrido-Atienza and J. Real, Asymptotic stability of nonlinear stochastic evolution equations,, \emph{Stoch. Anal. Appl.}, 21 (2003), 301. doi: 10.1081/SAP-120019288. Google Scholar

[2]

T. Caraballo and K. Liu, Exponential stability of mild solutions of stochastic partial differential equations with delays,, \emph{Stoch. Anal. Appl.}, 15 (1999), 743. doi: 10.1080/07362999908809633. Google Scholar

[3]

T. Caraballo, K. Liu and A. Truman, Stochastic functional partial differential equations: existence, uniqueness and asymptotic decay properties,, \emph{Proc. Roy. Soc. Lond. A}, 456 (2000), 1775. doi: 10.1098/rspa.2000.0586. Google Scholar

[4]

T. Caraballo, J. Real and L. Shaikhet, Method of Lyapunov functionals construction in stability of delay evolution equations,, \emph{J. Math. Anal. Appl.}, 334 (2007), 1130. doi: 10.1016/j.jmaa.2007.01.038. Google Scholar

[5]

H. Chen, Asymptotic behavior of stochastic two-dimensional Navier-Stokes equations with delays,, \emph{Proc. Indian Acad. Sci (Math. Sci)}, 122 (2012), 283. doi: 10.1007/s12044-012-0071-x. Google Scholar

[6]

V. Kolmanovskii and L. Shaikhet, A method of Lyapunov functionals construction for stochastic differential equations of neutral type,, \emph{Differentialniye uravneniya}, 31 (2002), 691. Google Scholar

[7]

V. Kolmanovskii and L. Shaikhet, Construction of Lyapunov functionals for stochastic hereditary systems: a survey of some recent results,, \emph{Mathematical and Computer Modelling}, 36 (1995), 1851. Google Scholar

[8]

J. Luo, Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays,, \emph{J. Math. anal. Appl.}, 342 (2008), 753. doi: 10.1016/j.jmaa.2007.11.019. Google Scholar

[9]

E. Pardoux, Equations aux dérivées partielles stochastiques nonlinéaires monotones,, Ph.D thesis, (1975). Google Scholar

[10]

G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions. Encyclopedia of mathematics and its applications,, Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223. Google Scholar

[11]

L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Difference Equations,, Springer, (2011). doi: 10.1007/978-0-85729-685-6. Google Scholar

[12]

L. Shaikhet, Modern state and development perspectives of Lyapunov functionals method in the stability theory of stochastic hereditary systems,, \emph{Theory of Stochastic Processes}, 2 (1996), 248. Google Scholar

[13]

L. Wan and J. Duan, Exponential stability of non-autonomous stochastic partial differential equations with finite memory,, \emph{Statistics and Probability Letters}, 78 (2008), 490. doi: 10.1016/j.spl.2007.08.003. Google Scholar

[14]

M. Wei and T. Zhang, Exponential stability for stochastic 2D-Navier-Stokes equations with time delay,, \emph{Appl. Math. J. Chinese Univ.}, 24 (2009), 493. Google Scholar

[1]

Hakima Bessaih, Benedetta Ferrario. Statistical properties of stochastic 2D Navier-Stokes equations from linear models. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2927-2947. doi: 10.3934/dcdsb.2016080

[2]

Yuri Bakhtin. Lyapunov exponents for stochastic differential equations with infinite memory and application to stochastic Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 697-709. doi: 10.3934/dcdsb.2006.6.697

[3]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295

[4]

Julia García-Luengo, Pedro Marín-Rubio, José Real. Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1603-1621. doi: 10.3934/cpaa.2015.14.1603

[5]

Hongyong Cui, Mirelson M. Freitas, José A. Langa. Squeezing and finite dimensionality of cocycle attractors for 2D stochastic Navier-Stokes equation with non-autonomous forcing. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1297-1324. doi: 10.3934/dcdsb.2018152

[6]

Lihuai Du, Ting Zhang. Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4745-4765. doi: 10.3934/dcds.2018209

[7]

Takeshi Taniguchi. The existence and decay estimates of the solutions to $3$D stochastic Navier-Stokes equations with additive noise in an exterior domain. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4323-4341. doi: 10.3934/dcds.2014.34.4323

[8]

G. Deugoué, T. Tachim Medjo. The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2593-2621. doi: 10.3934/cpaa.2018123

[9]

Kerem Uǧurlu. Continuity of cost functional and optimal feedback controls for the stochastic Navier Stokes equation in 2D. Communications on Pure & Applied Analysis, 2017, 16 (1) : 189-208. doi: 10.3934/cpaa.2017009

[10]

Yutaka Tsuzuki. Solvability of $p$-Laplacian parabolic logistic equations with constraints coupled with Navier-Stokes equations in 2D domains. Evolution Equations & Control Theory, 2014, 3 (1) : 191-206. doi: 10.3934/eect.2014.3.191

[11]

J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647

[12]

Songsong Lu, Hongqing Wu, Chengkui Zhong. Attractors for nonautonomous 2d Navier-Stokes equations with normal external forces. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 701-719. doi: 10.3934/dcds.2005.13.701

[13]

Ruihong Ji, Yongfu Wang. Mass concentration phenomenon to the 2D Cauchy problem of the compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1117-1133. doi: 10.3934/dcds.2019047

[14]

Min Zhu, Panpan Ren, Junping Li. Exponential stability of solutions for retarded stochastic differential equations without dissipativity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2923-2938. doi: 10.3934/dcdsb.2017157

[15]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515

[16]

Kai Liu. Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3915-3934. doi: 10.3934/dcdsb.2018117

[17]

Henri Schurz. Stochastic heat equations with cubic nonlinearity and additive space-time noise in 2D. Conference Publications, 2013, 2013 (special) : 673-684. doi: 10.3934/proc.2013.2013.673

[18]

Julia García-Luengo, Pedro Marín-Rubio, José Real. Regularity of pullback attractors and attraction in $H^1$ in arbitrarily large finite intervals for 2D Navier-Stokes equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 181-201. doi: 10.3934/dcds.2014.34.181

[19]

Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control & Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61

[20]

Kumarasamy Sakthivel, Sivaguru S. Sritharan. Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise. Evolution Equations & Control Theory, 2012, 1 (2) : 355-392. doi: 10.3934/eect.2012.1.355

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]